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An influence function asks how much a population answer moves when an infinitesimal fraction of its probability distribution is relocated to a specified point. It distinguishes the unbounded response of the mean from the bounded response of the median and the Huber location estimator. The normal-location influence curves and finite contamination bias of normal location below make that difference quantitative.

Let $T$ be a real-valued statistical functional, $F$ a reference probability measure, and ${\delta }_x$ the Dirac measure concentrated at $x$. Along the epsilon-contamination neighborhood path $F_{\epsilon ,x}=(1-\epsilon )F+\epsilon {\delta }_x$, define the right derivativeThe name influence curve refers to the same object. The derivative is in the signed direction ${\delta }_x-F$, so the calculation removes baseline mass as well as adding a Dirac measure. The definition is a limit for each fixed $x$; it does not automatically give a remainder uniform over arbitrarily distant $x$.^[ref]

The statistical functional must be defined on these contaminated probability measures, and the right derivative must exist and be finite. For a mean, a finite first absolute moment suffices. For a median, require a unique baseline median and a probability density function continuous and strictly positive there. For a smooth M-estimator, require an isolated population root, justified differentiation of its expected value, and a nonzero derivative with respect to the fitted parameter. These are specific conditions for the formulas below. Existence of the pointwise influence function alone proves neither an asymptotic linear representation nor pathwise differentiability of a statistical functional; each needs control of its own remainder and admissible statistical paths.

Consider any differentiable-in-quadratic-mean path with score function $g$. Put ${\delta }_t=\sqrt{f_t}-\sqrt{f}=(t/2)g\sqrt{f}+r_t$, where $\Vert r_t{\Vert }_2=o(|t|)$. The quadratic-mean to L1 density derivative follows fromBy the Cauchy-Schwarz inequality,Since $a$ is a bounded function, multiplying this L1 norm bound by $\Vert a{\Vert }_{\infty }$ provesThe last equality uses $P_{f}g=0$. The derivative is a bounded linear functional of $g\in L^2(P_f)$, so the required pathwise differentiability of a statistical functional holds, in particular relative to the statistical tangent set from part (b). Its derivative isFor the explicit bounded density tilts in part (b), this derivative is also obtained by direct integration, with no remainder term.